Digital Filter Arrangement for Compensating Group Velocity Dispersion in an Optical Transmission System

ABSTRACT

The present disclosure relates to a digital filter arrangement (DFA) for compensating group velocity dispersion (GVD) in an optical transmission system (OTS) wherein the DFA is configured to receive a sequence of samples of a digital input signal in the time domain in the form of consecutive blocks of size L. The DFA is configured to generate M discrete Fourier transforms of a current overlap block of a size N greater than the size L and of M−1 delayed versions of the current overlap block. The DFA is configured to filter the entries of the generated M discrete Fourier transforms to generate an output discrete Fourier transform with N entries, wherein the compensation filter is implemented by a delay network and a linear combination algorithm.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of International Application No.PCT/EP2020/068086, filed on Jun. 26, 2020, the disclosure of which ishereby incorporated by reference in its entirety.

TECHNICAL FIELD

The present disclosure relates to a digital filter arrangement (DFA) forcompensating group velocity dispersion (GVD) in an optical transmissionsystem (OTS), a transmitter comprising such a DFA and a receivercomprising such a DFA. The present disclosure further relates to anoptical transmission system (OTS) comprising such a transmitter and/orsuch a receiver and to a method for compensating GVD of an OTS usingsuch a DFA.

BACKGROUND

Transmission of modulated light over an optical fiber is impaired bygroup velocity dispersion (GVD). The group velocity describes the speedat which a light pulse propagates via the optical fiber. In the presenceof GVD, the group velocity of a light pulse varies over frequency, whichresults in pulse broadening and causes inter-symbol interference. In thefiber optics community, GVD is often referred to as chromatic dispersion(CD). The present disclosure is directed to compensating such GVDoccurring in optical fibers using digital filters respectively a digitalfilter arrangement (DFA). Thus, the present disclosure is in the fieldof DFAs for compensating GVD in an optical transmission system (OTS),wherein the OTS may comprise one or more optical fibers, across whichmodulated light is transmitted for an optical communication.

SUMMARY

Embodiments of the present disclosure base also on the followingconsiderations:

In single mode optical fibers (single mode fibers), GVD occurs as aconsequence of waveguide dispersion and material dispersion. In singlemode fibers, GVD is almost constant over the bandwidth of a singlechannel. As a consequence, the group delay may be approximately modeledas a linear function of the frequency. In modern optical systems it iscommon to modulate the amplitude, the phase and the polarization of thelight and to recover the modulated information by means of coherentoptical receivers.

A coherent optical receiver maps linearly the impinging light ontoelectrical signals. Since a single mode fiber carries two independentpolarizations (X polarization and Y polarization) and each polarization(X and Y) conveys a passband signal with two independent quadraturecomponents (I and Q), a coherent receiver maps the imping field onto twocomplex signals S1 and S2 (S1=X₁+j·X_(Q), S2=Y₁+j·Y_(Q), wherein j isthe imaginary unit). The complex baseband signals are sampled anddigitized to enable digital signal processing (DSP). A coherent receivermay post-compensate GVD by means of linear filters.

At the same way, a transmitter that is configured to modulate digitalsignals on the amplitude and the phase or, equivalently, on thequadrature components of the transmit signal may pre-compensate GVD bymeans of digital linear filters, independently of the detectiontechnique used at the receiver.

GVD may be compensated individually on each polarization because it doesnot involve any interaction between orthogonal polarizations. Since GVDis a unitary effect, i.e. energy-preserving effect, it may be post- orpre-compensated by linear all-pass filters.

Therefore, by applying digital linear all-pass filtering to the complexbaseband signals S1 and S2 (S1=P1·I+j·P1·Q, S2=P2·I+j·P2·Q, wherein j isthe imaginary unit) either at the transmitter or at the receiver, thenegative effect of GVD may be reverted, in principle without sufferingfrom any degradation of the signal quality.

Linear all-pass filters may be approximated by digital finite impulseresponse (FIR) filters. FIR filters are desirable because they areinherently stable, also in finite-precision arithmetic, and may berealized via feed-forward structures.

However, in practice, very long FIR filters, i.e. filters with a greatsize, are required to compensate GVD. The filter length growsproportionally to the transmission distance and to the square of thesymbol rate, and, the implementation complexity grows with the filterlength. The terms “filter size” and “filter length” may be used assynonyms.

Therefore, whereas in theory GVD may be compensated perfectly by usingdigital linear filters, the complexity of digital GVD compensation foroptical transmission systems, such as optical coherent transmissionsystems, grows rapidly with the transmission distance and the symbolrate, and becomes challenging in the case of long-haul, high-rateapplications.

In view of the above-mentioned problems and disadvantages, embodimentsof the present disclosure aim to reduce the complexity of GVDcompensation using a digital filter arrangement (DFA). An objective isto provide a digital filter arrangement (DFA) for compensating GVD in anoptical transmission system (OTS) with a reduced complexity of the GVDcompensation.

The objective is achieved by the embodiments of the disclosure asdescribed in the enclosed independent claims. Advantageousimplementations of the embodiments of the disclosure are further definedin the dependent claims.

A first aspect of the present disclosure provides a digital filterarrangement (DFA) for compensating group velocity dispersion (GVD) in anoptical transmission system (OTS), wherein the DFA is configured toreceive a sequence of samples of a digital input signal in the timedomain in the form of consecutive blocks of size L, wherein each blockcomprises L consecutive samples of the digital input signal. The DFA isconfigured to generate M discrete Fourier transforms of a currentoverlap block of a size N greater than the size L and of M−1 delayedversions of the current overlap block, by using M discrete Fouriertransform (DFT) filters, wherein each generated discrete Fouriertransform is of the size N and comprises N entries, the current overlapblock comprises the samples of a current block and the N-L lastconsecutive samples of a directly previous block that was received bythe DFA directly before the current block, and each DFT filter of the MDFT filters is implemented by a DFT algorithm, in particular by a fastFourier transform, FFT, algorithm, of a size Γ smaller than the size Nand by an interpolation algorithm.

The DFA is configured to filter, by a compensation filter, the entriesof the generated M discrete Fourier transforms to generate an outputdiscrete Fourier transform with N entries, wherein the compensationfilter is implemented by a delay network and a linear combinationalgorithm.

Thus, the present disclosure proposes a filtering using a DFA, whereinthe samples of a current block of a digital signal in the time domainare transformed into the frequency domain and then filtered, using acompensation filter, in the frequency domain. In other words, the DFAaccording to the first aspect is based on frequency domain filtering.The DFA may be configured to generate for a current block M discreteFourier transforms of a respective current overlap block and of M−1delayed versions of the current overlap block by using the M DFTfilters.

As a result of the frequency domain filtering the complexity of the DFAaccording to the first aspect is reduced, because a frequency domainimplementation of filtering comprises a lower complexity compared totime domain implementations. Namely, in contrast to a time domainimplementation of filtering, in a frequency domain implementation offiltering the number of operations per sample grows logarithmicallyinstead of linearly with the filter size. The term “length” may be usedas a synonym for the term “size”. In addition, as a result of using theM DFT filters the complexity of the DFA is further reduced. That is, thetransformation of the samples from the time domain to the frequencydomain using the M DFT filters requires a lower complexity compared toperforming a normal DFT algorithm. Namely, for transforming a sequenceof N samples from the time domain to the frequency domain a DFTalgorithm of size N and, thus, a DFT filter of size N is normallyrequired. In contrast thereto, the present disclosure proposes to use MDFT filters, wherein each DFT filter is implemented by a DFT algorithmof a size Γ smaller than the size N and by an interpolation algorithm.That is, the present disclosure proposes to reduce the size of the DFTalgorithm from N to F and to compensate this reduction by additionallyperforming an interpolation algorithm.

Therefore, the present disclosure proposes with the DFA according to thefirst aspect a DFA with a reduced complexity by performing the filteringin the frequency domain and reducing the complexity of thetransformation of the samples from the time domain to the frequencydomain for the frequency domain filtering.

As outlined above, the DFT algorithm of the size Γ may optionally be afast Fourier transform algorithm (FFT algorithm) of the size F.

The DFT algorithm of size Γ may optionally be performed by using asquare DFT matrix of size F. That is, the transformation of samples fromthe time domain to the frequency domain may optionally be performed byusing a DFT matrix of size F.

The discrete Fourier transform of samples (e.g. the samples of a currentoverlap block) corresponds to the transformation output respectivelytransformation result in the frequency domain of a DFT filter used fortransforming the samples from the time domain, which are input to theDFT filter as samples in the time domain, to the frequency domain. Thatis, the discrete Fourier transform of samples corresponds to thetransformation output of the DFT algorithm used for transforming thesamples from the time domain to the frequency result.

The samples in the time domain (e.g. the samples of a current block orof a current overlap block) may correspond to a matrix comprising thesamples in the time domain as entries. A discrete Fourier transform maycorrespond to a matrix with entries in the frequency domain. A vector isa special case of a matrix, namely a matrix with only one row or withonly one column.

Therefore, for generating, on the basis of a matrix comprising thesamples of the digital signal in the time domain as entries (such as thesamples of a current block or current overlap block), the matrix of thecorresponding discrete Fourier transform, the corresponding DFT filtermay be used. The corresponding DFT filter may optionally be implementedby a square DFT matrix of the size Γ (implementing the DFT algorithm ofthe size F) and an interpolation algorithm.

The N entries of a discrete Fourier transform correspond to Fouriercoefficients.

The output discrete Fourier transform is compensated with respect to theGVD.

According to an implementation form, the DFA is based on frequencydomain filtering according to an overlap-save method or on frequencydomain filtering according to an overlap-add method.

The DFA may be implemented by hardware and/or software.

As outlined already above, the term “group velocity dispersion (GVD)”may also be referred to as “chromatic dispersion (CD)”.

In an implementation form of the first aspect, the DFA is configured togenerate, on the basis of the output discrete Fourier transform, anoutput block of size N by using an inverse discrete Fouriertransformation (IDFT) filter.

In an implementation form of the first aspect, the DFA is configured togenerate an output block of the size L on the basis of the output blockof the size N, by using an overlap-add or an overlap-save method.

Thus, the size of the output block may be reduced from the size N to thesize L.

The overlap-save method may also be referred to by the terms“overlap-discard method” and “overlap-scrap method”.

In particular, the size L of each block is the product of M and Δ(L=M·Δ), wherein M and A are each a positive integer.

Each block may be made of M subblocks of the size Δ.

In particular, the size N of the current overlap block is the product ofΔ with the sum of M and m (N=Δ·(M+m)), wherein m is a positive integer,and wherein the product of Δ and m (Δ·m) corresponds to the number oflast consecutive samples in the sequence of samples of the directlyprevious block.

The integer m corresponds to the number of last consecutive subblocks ofthe size Δ of the directly previous block.

In an implementation form of the first aspect, the DFA is configured togenerate, on the basis of the current block, the current overlap blockof the size N greater than the size L, generate the M−1 delayed versionsof the current overlap block, and jointly approximate, on the basis ofthe current overlap block and the M−1 delayed versions of the currentoverlap block, the M discrete Fourier transforms by using the M DFTfilters.

Thus, the overlapping for generating the current overlap block and thedelaying for generating the M−1 delayed versions of the current overlapblock may be performed in the time domain.

Optionally, the current overlap block and the M−1 delayed versions ofthe current overlap block may be delayed, wherein the relative delaysamong the blocks are not changed. In other words, the current overlapblock may optionally comprise a base latency and the M−1 delayedversions of the current overlap block each may comprise additionallatency.

The current overlap block may be generated by prepending to the currentblock the last N-L samples of the directly previous block.

In an implementation form of the first aspect, the DFA is configured togenerate the M−1 delayed versions of the current overlap block byprogressively delaying the current overlap block in steps of Δ samples.

The first delayed version of the M−1 delayed versions of the currentoverlap block may correspond to the current overlap block delayed by Δsamples, the second delayed version of the current overlap block maycorrespond to the current overlap block delayed by 2Δ samples, and soon. That is, the second delayed version may correspond to the firstdelayed version delayed by Δ samples, and so on. Therefore, the M−1^(th)delayed version may correspond to the current overlap block delayed by(M−1)·Δ samples.

In an implementation form of the first aspect, the DFA is configured tosplit the current block into M subblocks each of a size Δ, wherein eachsubblock comprises Δ consecutive samples of the current block, and tozero-pad each subblock to a zero-padded sequence of the size Γcomprising F samples, wherein each zero-padded sequence comprises the Δsamples of the corresponding subblock and Γ-Δ zeros. Further, the DFAmay be configured to jointly approximate, on the basis of the Mzero-padded sequences, M further discrete Fourier transforms by usingthe M DFT filters. Furthermore, the DFA may be configured to generatethe M discrete Fourier transforms by delaying the M further discreteFourier transforms, and by aligning and adding one or more of the Mdelayed further discrete Fourier transforms and one or more of the Mfurther discrete Fourier transforms.

The M further discrete Fourier transforms may be referred to as“input-pruned discrete Fourier transforms”.

Each of the M further discrete Fourier transforms may correspond to adiscrete Fourier transform with only a subset of non-zero input points.

In an implementation form of the first aspect, the DFA is configured toperform the alignment by performing a symbol-wise multiplication of oneor more delayed further discrete Fourier transforms and/or one or morefurther discrete Fourier transforms by a rotation vector.

The symbol-wise multiplication may also be referred to as a point-wisemultiplication.

The DFA may be configured to perform the alignment by performing asymbol-wise multiplication of one or more delayed further discreteFourier transforms and/or one or more further discrete Fouriertransforms by a rotation vector whose elements are powers of theelements of a base rotation vector.

${{\rho\_}1_{M + m}{\rho\_}1_{M + m}}\overset{\bigtriangleup}{=}{\exp\left( {j{\frac{2\pi}{M + m}\left\lbrack {0,1,\ldots,{N - 1}} \right\rbrack}} \right)}$

The base rotation vector may be as follows:

${{{\rho\_}1_{M + m}{\rho\_}1_{M + m}}\overset{\bigtriangleup}{=}{\exp\left( {j{\frac{2\pi}{M + m}\left\lbrack {0,1,\ldots,{N - 1}} \right\rbrack}} \right)}},$

wherein j is the imaginary unit.

In an implementation form of the first aspect, the DFA is configured togroup the A samples of each subblock into two parts of samples,optionally into two equal parts of samples, to carry out thezero-padding of each subblock to a zero-padded sequence of the size Γ byadding Γ-Δ zeros between the two parts of samples, and to jointlyapproximate, on the basis of the M zero-padded sequences, the M furtherdiscrete Fourier transforms by using the M DFT filters, wherein each DFTfilter is implemented by a FFT algorithm of the size Γ and aninterpolation algorithm.

The DFA may be configured to group the Δ samples of each subblock into afirst part (x+) of samples (first part comprising consecutivecoefficients) and a second part (x−) of samples (second part comprisingconsecutive coefficients). The number of samples of the first and secondpart may optionally be the same. That is, the DFA may optionally beconfigured to group the Δ samples of each subblock into two equal partsof samples.

The first and second part may be arranged in the respective subblock,such that in the sequence of consecutive samples of the respectivesubblock the samples of the second part are arranged previous to thesamples of the first part ([x−, x+]. That is, the samples of the secondpart may be arranged at the beginning and the samples of the first partmay be arranged at the end of the respective subblock. Thus, the samplesof the first part (x+) may follow the samples of the second part (x−) inthe sequence of consecutive samples of the respective subblock.

In an implementation form of the first aspect, the zero-padding of eachsubblock generates a zero-padded sequence of the size Γ by adding Γ-Δzeros between the first part of samples and the second part of samplesof the respective subblock, wherein the positions of the first part andsecond part are swapped. That is, in the zero-padded sequence ofconsecutive samples, generated on the basis of each subblock, the firstand second part of the respective subblock and the Γ-Δ zeros may bearranged, such that in the zero-padded sequence the samples of the firstpart (x+) are arranged previous to the samples of the second part (x−),wherein the Γ-Δ zeros are arranged between the first and second part([x+, 0, . . . , 0, x−]). That is, the samples of the second part mayfollow the samples of the first part in the zero-padded sequence ofconsecutive samples of the respective subblock, wherein the Γ-Δ zerosare arranged between the first and second part. Thus, afterzero-padding, the samples of the first part (x+) may be arranged at thebeginning of the zero-padded sequence and the samples of the second part(x−) may be arranged at the end of the zero-padded sequence, wherein theΓ-Δ zeros are arranged between these two parts. The first part (x+) maybe referred to as the front part (x_front) and the second part (x−) maybe referred to as the tail part (x tail), wherein the terms “front” and“tail” refer to the positions of the two parts in the zero-paddedsequence after the optional reordering described above.

Thus, the zero-padding of each subblock may entail a reordering of thetwo parts of samples of the respective subblock.

In an implementation form of the first aspect, the DFT is configured tojointly approximate on the basis of the M zero-padded sequences the Mfurther discrete Fourier transforms by transforming each zero-paddedsequence by an FFT algorithm of the size Γ to a discrete Fouriertransform of the size F, interpolating the Γ samples of each discreteFourier transform of the size Γ to N samples of another discrete Fouriertransform of the size N by a low-pass filter, and performing asymbol-wise multiplication of the samples of each other discrete Fouriertransform by a rotation vector to obtain the respective further discreteFourier transform.

The symbol-wise multiplication of the samples of each other discreteFourier transform by the rotation vector corresponds to a shift in thetime domain of the non-zero samples of the respective zero-paddedsequence.

${{\rho\_}2_{M + m}{\rho\_}2_{M + m}}\overset{\bigtriangleup}{=}{\exp\left( {j{\frac{\pi}{M + m}\left\lbrack {0,1,\ldots,{N - 1}} \right\rbrack}} \right)}$

The rotation vector may be as follows:

${{{\rho\_}2_{M + m}{\rho\_}2_{M + m}}\overset{\bigtriangleup}{=}{\exp\left( {j{\frac{\pi}{M + m}\left\lbrack {0,1,\ldots,{N - 1}} \right\rbrack}} \right)}},$

wherein j is the imaginary unit.

The interpolation may be implemented by using a polyphase finite impulseresponse (FIR) filter, wherein the input of the polyphase FIR filter isregarded as periodic.

In an implementation form of the first aspect, the compensation filtercomprises N subfilters, wherein each subfilter is implemented by a delaynetwork and a linear combination algorithm, and the DFA is configured toperform a filtering on the k^(th) entry of one or more of the generatedM discrete Fourier transforms by using the k^(th) subfilter to generatethe k^(th) entry of the output Fourier transform, wherein each discreteFourier transform comprises N entries. (k is an integer between 1 and N,k=1, . . . , N).

Each subfilter may comprise a single output and one or more inputs. Inan implementation form, each subfilter may comprise a single output andmultiple inputs. That is, each subfilter may be a multiple-inputsingle-output filter.

In an implementation form of the first aspect, the compensation filter,in particular the k^(th) subfilter, is configured to generate the k^(th)entry of the output Fourier transform as a linear combination of thek^(th) entry of one or more first discrete Fourier transforms of the Mdiscrete Fourier transforms, and/or the k^(th) entry of one or moredelayed versions of one or more second discrete Fourier transforms ofthe M discrete Fourier transforms.

In other words, the compensation filter, in particular the k^(th)subfilter, may be configured to generate the k^(th) entry of the outputFourier transform as a linear combination of the k^(th) entry of one ormore discrete Fourier transforms (may be referred to as “one or morefirst discrete Fourier transforms”) of the M discrete Fouriertransforms, and/or the k^(th) entry of one or more delayed versions ofone or more discrete Fourier transforms (may be referred to as “one ormore second discrete Fourier transforms”) of the M discrete Fouriertransforms.

The one or more delayed versions of a discrete Fourier transform maydiffer to each other by a delay of multiples of L samples in the timedomain or a delay of multiples of blocks of the respective discreteFourier transform in the frequency domain. That is, a delayed version ofa discrete Fourier transform may correspond to the discrete Fouriertransform delayed by a delay of multiples of blocks of the discreteFourier transform in the frequency domain.

In an implementation form of the first aspect, some or all of the one ormore first discrete Fourier transforms and of the one or more seconddiscrete Fourier transforms may correspond to each other.

In an implementation form of the first aspect, the compensation filter,in particular the k^(th) subfilter, is configured to perform the linearcombination by weighting the k^(th) entry of the one or more firstdiscrete Fourier transforms, and/or the k^(th) entry of the one or moredelayed versions of the one or more second discrete Fourier transformswith a respective coefficient.

The compensation filter, in particular the k^(th) subfilter, may beconfigured to linearly combine the k^(th) entry of the one or more firstdiscrete Fourier transforms and/or the k^(th) entry of the one or moredelayed versions of the one or more second discrete Fourier transformsby using one or more corresponding coefficients of the linearcombination algorithm.

In an implementation form of the first aspect, the compensation filter,in particular the k^(th) subfilter, is configured to generate the k^(th)entry of each of the one or more delayed versions of each of the one ormore second discrete Fourier transforms by delaying the k^(th) entry ofthe respective second discrete Fourier transform using one or moreinteger delays.

The k^(th) subfilter may be configured to generate, using the respectivedelay network, the k^(th) entry of each of the one or more delayedversions of each of the one or more second discrete Fourier transformsby delaying the k^(th) entry of the respective second discrete Fouriertransform using one or more integer delays.

In an implementation form of the first aspect, the DFA is configured tooptimize, by an optimization method, the one or more coefficients and/orthe one or more integer delays with regard to the GVD compensation.

In order to achieve the DFA according to the first aspect of the presentdisclosure, some or all of the implementation forms and optionalfeatures of the first aspect, as described above, may be combined witheach other.

A second aspect of the present disclosure provides a transmitter for anoptical transmission system (OTS), wherein the transmitter is configuredto transmit, on the basis of a digital signal, a modulated light signalvia one or more optical fibers to a receiver and wherein the transmitteris configured to pre-compensate group velocity dispersion (GVD) of themodulated light signal on the basis of the digital signal using adigital filter arrangement (DFA) according to the first aspect or any ofits implementation forms.

The transmitter comprises the DFA according to the first aspect or anyof its implementation forms and, thus, is configured to pre-compensateGVD of the modulated light signal on the basis of the digital signalusing the DFA.

The one or more optical fibers may correspond to one or more single modefibers.

The transmitter of the second aspect and its implementation forms andoptional features achieve the same advantages as the DFA of the firstaspect and its respective implementation forms and respective optionalfeatures.

A third aspect of the present disclosure provides a receiver for anoptical transmission system (OTS), wherein the receiver is configured toreceive a modulated light signal via one or more optical fibers from atransmitter and to convert the received modulated light signal into adigital signal, and wherein the receiver is configured to compensategroup velocity dispersion (GVD) of the received modulated light signalon the basis of the digital signal using a digital filter arrangement(DFA) according to the first aspect or any of its implementation forms.

The receiver comprises the DFA according to the first aspect or any ofits implementation forms and, thus, is configured to compensate GVD ofthe received modulated light signal on the basis of the digital signalusing the DFA.

The one or more optical fibers may correspond to one or more single modefibers.

In case the optical fiber is a single mode fiber, the receiver may beconfigured to map the received modulated light signal onto two complexsignals S1 and S2 with two independent quadrature components I and Q(S1=X₁+j·X_(Q), S2=Y₁+j·Y_(Q), wherein j is the imaginary unit). Namely,a single mode fiber carries two independent polarizations (Xpolarization and Y polarization) and each polarization (X and Y) conveysa passband signal with two independent quadrature components I and Q.

In an implementation form of the third aspect, the receiver may be acoherent-detection receiver.

The receiver of the third aspect and its implementation forms andoptional features achieve the same advantages as the DFA of the firstaspect and its respective implementation forms and respective optionalfeatures.

In order to achieve the receiver according to the third aspect of thepresent disclosure, some or all of the implementation forms and optionalfeatures of the third aspect, as described above, may be combined witheach other.

A fourth aspect of the present disclosure provides an opticaltransmission system (OTS), wherein the OTS comprises one or more opticalfibers, a transmitter and a receiver. The transmitter is configured totransmit, on the basis of a first digital signal, a modulated lightsignal via the one or more optical fibers to the receiver. The receiveris configured to receive the modulated light signal via the one or moreoptical fibers from the transmitter and convert the received modulatedlight signal into a second digital signal. The transmitter is atransmitter according to the second aspect or any of its implementationforms, which is configured to pre-compensate group velocity dispersion(GVD) of the modulated light signal on the basis of the first digitalsignal using the digital filter arrangement (DFA) according to the firstaspect or any of its implementation forms. Alternatively oradditionally, the receiver is a receiver according to the third aspector any of its implementation forms, which is configured to compensateGVD of the received modulated light signal on the basis of the seconddigital signal using the DFA according to the first aspect or any of itsimplementation forms.

The one or more optical fibers may correspond to one or more single modefibers.

In an implementation form of the fourth aspect, the receiver may be acoherent-detection receiver. In particular, the receiver may be acoherent-detection receiver, in case of group velocity dispersion (GVD)post-compensation, that is in case of no pre-compensation by thetransmitter but compensation by the receiver.

In an implementation form of the fourth aspect, the receiver may be acoherent-detection receiver or a direct detection receiver, in case ofGVD pre-compensation by the transmitter.

The OTS of the fourth aspect and its implementation forms and optionalfeatures achieve the same advantages as the DFA of the first aspect andits respective implementation forms and respective optional features.

In order to achieve the OTS of the fourth aspect of the presentdisclosure, some or all of the implementation forms and optionalfeatures of the fourth aspect, as described above, may be combined witheach other.

A fifth aspect of the present disclosure provides a method forcompensating group velocity dispersion (GVD) of an optical transmissionsystem (OTS) using a digital filter arrangement (DFA) according to thefirst aspect or any of its implementation forms, wherein the methodcomprises the step of receiving a sequence of samples of a digital inputsignal in the time domain in the form of consecutive blocks of size L,wherein each block comprises L consecutive samples of the digital inputsignal. The method further comprises the step of generating M discreteFourier transforms of a current overlap block of a size N greater thanthe size L and of M−1 delayed versions of the current overlap block byusing M discrete Fourier transform (DFT) filters. Each generateddiscrete Fourier transform is of the size N and comprises N entries. Thecurrent overlap block comprises the samples of a current block and theN-L last consecutive samples of a directly previous block that wasreceived by the DFA directly before the current block. Each DFT filterof the M DFT filters is implemented by a DFT algorithm, in particular bya fast Fourier transform (FFT) algorithm, of a size Γ smaller than thesize N and by an interpolation algorithm. The method further comprisesthe step of filtering, by a compensation filter, the entries of thegenerated M discrete Fourier transforms to generate an output discreteFourier transform with N entries, wherein the compensation filter isimplemented by a delay network and a linear combination algorithm.

In an implementation form of the fifth aspect, the method comprises thestep of generating, on the basis of the output discrete Fouriertransform, an output block of size N by using an inverse discreteFourier transformation (IDFT) filter.

In an implementation form of the fifth aspect, the method comprises thestep of generating an output block of the size L on the basis of theoutput block of the size N, by using an overlap-add or an overlap-savemethod.

In an implementation form of the fifth aspect, the method comprises thesteps of generating, on the basis of the current block, the currentoverlap block of the size N greater than the size L, generating the M−1delayed versions of the current overlap block, and jointlyapproximating, on the basis of the current overlap block and the M−1delayed versions of the current overlap block, the M discrete Fouriertransforms by using the M DFT filters.

In an implementation form of the fifth aspect, the method comprises thestep of generating the M−1 delayed versions of the current overlap blockby progressively delaying the current overlap block in steps of Δsamples.

In an implementation form of the fifth aspect, the method comprises thestep of splitting the current block into M subblocks each of a size Δ,wherein each subblock comprises Δ consecutive samples of the currentblock, and the step of zero-padding each subblock to a zero-paddedsequence of the size Γ comprising F samples, wherein each zero-paddedsequence comprises the Δ samples of the corresponding subblock and Γ-Δzeros. Further, the method may comprise the step of jointlyapproximating, on the basis of the M zero-padded sequences, M furtherdiscrete Fourier transforms by using the M DFT filters. Furthermore, themethod may comprise the step of generating the M discrete Fouriertransforms by delaying the M further discrete Fourier transforms, and byaligning and adding one or more of the M delayed further discreteFourier transforms and one or more of the M further discrete Fouriertransforms.

In an implementation form of the fifth aspect, the method comprises thestep of performing the alignment by performing a symbol-wisemultiplication of one or more delayed further discrete Fouriertransforms and/or one or more further discrete Fourier transforms by arotation vector.

In an implementation form of the fifth aspect, the method comprises thestep of grouping the Δ samples of each subblock into two parts ofsamples, optionally into two equal parts of samples, the step ofcarrying out the zero-padding of each subblock to a zero-padded sequenceof the size Γ by adding Γ-Δ zeros between the two parts of samples, andthe step of jointly approximating, on the basis of the M zero-paddedsequences, the M further discrete Fourier transforms by using the M DFTfilters, wherein each DFT filter is implemented by a FFT algorithm ofthe size Γ and an interpolation algorithm.

In an implementation form of the fifth aspect, the method comprises thestep of jointly approximating on the basis of the M zero-paddedsequences the M further discrete Fourier transforms by transforming eachzero-padded sequence by an FFT algorithm of the size F to a discreteFourier transform of the size F, interpolating the Γ samples of eachdiscrete Fourier transform of the size Γ to N samples of anotherdiscrete Fourier transform of the size N by a low-pass filter, andperforming a symbol-wise multiplication of the samples of each otherdiscrete Fourier transform by a rotation vector to obtain the respectivefurther discrete Fourier transform.

In an implementation form of the fifth aspect, the compensation filtercomprises N subfilters, wherein each subfilter is implemented by a delaynetwork and a linear combination algorithm, and the method comprises thestep of performing a filtering on the k^(th) entry of one or more of thegenerated M discrete Fourier transforms by using the k^(th) subfilter togenerate the k^(th) entry of the output Fourier transform, wherein eachdiscrete Fourier transform comprises N entries. (k is an integer between1 and N, k=1, . . . , N)

In an implementation form of the fifth aspect, the method comprises thestep of generating, by the compensation filter, in particular by thek^(th) subfilter, the k^(th) entry of the output Fourier transform as alinear combination of the k^(th) entry of one or more first discreteFourier transforms of the M discrete Fourier transforms, and/or thek^(th) entry of one or more delayed versions of one or more seconddiscrete Fourier transforms of the M discrete Fourier transforms.

In an implementation form of the fifth aspect, the method comprises thestep of performing, by the compensation filter, in particular by thek^(th) subfilter, the linear combination by weighting the k^(th) entryof the one or more first discrete Fourier transforms, and/or the k^(th)entry of the one or more delayed versions of the one or more seconddiscrete Fourier transforms with a respective coefficient.

In an implementation form of the fifth aspect, the method comprises thestep of generating, by the compensation filter, in particular by thek^(th) subfilter, the k^(th) entry of each of the one or more delayedversions of each of the one or more second discrete Fourier transformsby delaying the k^(th) entry of the respective second discrete Fouriertransform using one or more integer delays.

In an implementation form of the fifth aspect, the method comprises thestep of optimizing, by an optimization method, the one or morecoefficients and/or the one or more integer delays with regard to theGVD compensation.

The method of the fifth aspect and its implementation forms and optionalfeatures achieve the same advantages as the DFA of the first aspect andits respective implementation forms and respective optional features.

The implementation forms and optional features of the DFA according tothe first aspect are correspondingly valid for the method according tothe fifth aspect.

In order to achieve the method according to the fifth aspect of thepresent disclosure, some or all of the implementation forms and optionalfeatures of the fifth aspect, as described above, may be combined witheach other.

A sixth aspect of the present disclosure provides a computer programcomprising program code for performing when implemented on a processor,a method according to the fifth aspect or any of its implementationforms.

A seventh aspect of the present disclosure provides a computer programcomprising a program code for performing the method according to thefifth aspect or any of its implementation forms.

An eighth aspect of the present disclosure provides a computercomprising a memory and a processor, which are configured to store andexecute program code to perform the method according to the fifth aspector any of its implementation forms.

A ninth aspect of the present disclosure provides a non-transitorystorage medium storing executable program code which, when executed by aprocessor, causes the method according to the fifth aspect or any of itsimplementation forms to be performed.

It has to be noted that all devices, elements, units and means describedin the present application could be implemented in the software orhardware elements or any kind of combination thereof. All steps whichare performed by the various entities described in the presentapplication as well as the functionalities described to be performed bythe various entities are intended to mean that the respective entity isadapted to or configured to perform the respective steps andfunctionalities. Even if, in the following description of specificembodiments, a specific functionality or step to be performed byexternal entities is not reflected in the description of a specificdetailed element of that entity which performs that specific step orfunctionality, it should be clear for a skilled person that thesemethods and functionalities can be implemented in respective software orhardware elements, or any kind of combination thereof.

BRIEF DESCRIPTION OF THE DRAWINGS

The above described aspects and implementation forms will be explainedin the following description of specific embodiments in relation to theenclosed drawings, in which:

FIG. 1 shows a digital filter arrangement (DFA) according to anembodiment;

FIG. 2 shows a digital filter arrangement (DFA) according to anembodiment;

FIG. 3 shows an example of a transformation from the time domain to thefrequency domain and a delaying in the frequency domain used in adigital filter arrangement (DFA) according to an embodiment;

FIG. 4 shows an example of an aligning and adding of discrete Fouriertransforms in the frequency domain;

FIG. 5 shows the generation of the M further discrete Fourier transformsshown in FIGS. 3 and 6 by a digital filter arrangement (DFA) accordingto an embodiment;

FIG. 6 shows a generation of M discrete Fourier transforms of a currentoverlap block of a size N greater than the size L and of M−1 delayedversions of the current overlap block by using M discrete Fouriertransform (DFT) filters of a digital filter arrangement (DFA) accordingto an embodiment, wherein M equals three (M=3), L equals three times Δsamples (L=3·Δ), A is a generic positive integer number and N equalsfour times Δ samples (N=4·Δ);

FIG. 7A shows a transmitter according to an embodiment;

FIG. 7B shows a receiver according to an embodiment; and

FIG. 8 shows an optical transmission system (OTS) according to anembodiment.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

FIG. 1 shows a digital filter arrangement (DFA) according to anembodiment of the present disclosure.

The above description of the DFA according to the first aspect and itsimplementation forms is correspondingly valid for the DFA of FIG. 1 .

The DFA 1 of FIG. 1 comprises two parts 2, 3. The first part 2 of theDFA 1 is configured for transforming samples of a sequence of samples ofa digital input signal in the time domain from the time domain to thefrequency domain. The second part 3 of the DFA 1 is configured toperform a filtering in the frequency domain of the transformed samplesfor compensating group velocity dispersion (GVD).

The first part 2 of the DFA 1 comprises M discrete Fourier transform(DFT) filters 2 a. The first part 2 of the DFA may be configured toreceive the sequence of samples of the digital input signal in the timedomain in the form of consecutive blocks S respectively S(μ) of size L,wherein each block comprises L consecutive samples of the digital inputsignal. The “μ” corresponds to a time point, such as the time point ofthe clock of a processor for executing the function of the DFA 1.

The first part 2 of the DFA 1 may be configured to generate M discreteFourier transforms X₁, X₂, . . . , X_(M) of a current overlap block of asize N greater than the size L and of M−1 delayed versions of thecurrent overlap block by using the M DFT filters 2 a. In other words,the first part 2 of the DFA may be configured to perform for a currentblock a transformation from the time domain to the frequency domainusing the M DFT filters 2 a generating M discrete Fourier transforms X₁,X₂, . . . , X_(M) for a current overlap block of a size N greater thanthe size L and M−1 delayed versions of the current overlap block,wherein the current overlap block is based on the current block. The Mdiscrete Fourier transforms X₁, X₂, . . . , X_(M) are the results of thetransformation from the time domain to the frequency domain.

Each DFT filter of the M DFT filters 2 a is implemented by a DFTalgorithm of a size F smaller than the size N and by an interpolationalgorithm. The DFT algorithm of the size Γ may be a fast Fouriertransform (FFT) algorithm of the size N.

The size L of each block S respectively S(μ) is the product of M and Δ(L=M·Δ), wherein M and Δ are each a positive integer. The size N of thecurrent overlap block is the product of Δ with the sum of M and m(N=Δ·(M+m)), wherein m is a positive integer and the product of Δ and m(Δ·m) corresponds to the number of last consecutive samples in thesequence of samples of the directly previous block.

Each generated discrete Fourier transform of the M discrete Fouriertransforms X₁, X₂, . . . , X_(M) is of the size N and comprises Nentries. The entries may also be referred to as Fourier coefficients orFourier components. The current overlap block comprises the samples ofthe current block and the N-L last consecutive samples of a directlyprevious block that was received by the first part 2 of the DFA 1directly before the current block.

The second part 3 of the DFA 1 corresponds to a compensation filter. Thecompensation filter 3 is configured to filter the entries of thegenerated M discrete Fourier transforms X₁, X₂, . . . , X_(M) togenerate an output discrete Fourier transform Y with N entries Y₁, Y₂, .. . , Y_(N). The output discrete Fourier transform Y is compensated withrespect to the GVD. The compensation filter 3 is implemented by a delaynetwork 3 a and a linear combination algorithm 3 b.

As a result of the frequency domain filtering using the compensationfilter 3 the complexity of the DFA 1 is reduced, because a frequencydomain implementation of filtering comprises a lower complexity comparedto time domain implementations. Namely, in contrast to a time domainimplementation of filtering, in a frequency domain implementation offiltering the number of operations per sample grows logarithmicallyinstead of linearly with the filter size. In addition, as a result ofusing the DFT filters 2 a the complexity of the DFA is further reduced.That is, the transformation of the samples from the time domain to thefrequency domain using the DFT filters 2 a requires a lower complexitycompared to performing a normal DFT algorithm. Namely, for transforminga sequence of N samples from the time domain to the frequency domain aDFT algorithm of size N and, thus, a DFT filter of size N is normallyrequired. In contrast thereto, the present disclosure proposes to usethe M DFT filters 2 a, wherein each DFT filter 2 a is implemented by aDFT algorithm of a size Γ smaller than the size N and by aninterpolation algorithm. That is, the present disclosure proposes toreduce the size of the DFT algorithm from N to F and to compensate thisreduction by additionally performing an interpolation algorithm.

As outlined already above, the DFA 1 may be used in a receiver and/or atransmitter of an optical transmission system. Thus, the sequence ofinput blocks S may correspond to the complex baseband signal that shallbe pre-compensated at the transmitter or post-compensated at thereceiver in case of using one or more single-mode fibers forcommunication in an optical transmission system, in which thetransmitter respectively receiver is provided. Since, as discussedabove, GVD compensation may be applied separately to orthogonalpolarizations, only one polarization is considered. The filtering usingthe DFA 1 for GVD compensation may also be applied to the secondpolarization, if necessary.

The DFA 1 may be implemented by hardware and/or software.

The DFA 1 according to FIG. 1 allows the use of any number of DFTfilters 2 a and the implementation of any overlap ratio, wherein theoverlap ratio is defined as

$\frac{m}{\left( {M + m} \right)}$

(M and m are each a positive integer). This is advantageous, becausearbitrarily chosen low overlap ratios are possible by increasing thenumber M of DFT filters 2 a. Due to the efficient joint approximation(joint computation) of the M discrete Fourier transforms X₁, X₂, . . . ,X_(M) using the M DFT filters 2 a, this results in complexity savings.The size N of the used M discrete Fourier transforms X₁, X₂, . . . ,X_(M) may be reduced by increasing their number M. Due to the efficientjoint computation of the M discrete Fourier transforms X₁, X₂, . . . ,X_(M) using the M DFT filters 2 a, this results in complexity savings atequal or better group velocity dispersion (GVD) tolerance.

According to an implementation form, the M DFT filters 2 a and, thus,the M discrete Fourier transforms X₁, X₂, . . . , X_(M) may beprogressively activated on demand according to the requested GVDtolerance in order to save complexity and power consumption over shorterlinks.

FIG. 2 shows a digital filter arrangement (DFA) according to anembodiment of the present disclosure.

The above description of the DFA according to the first aspect and itsimplementation forms is correspondingly valid for the DFA of FIG. 2 .

The DFA 1 of FIG. 2 corresponds to the DFA 1 of FIG. 1 , wherein FIG. 2shows an implementation form of the first part 2 and the compensationfilter 3 (second part) of the DFA 1 as well as optional additionalfeatures of the DFA 1 not shown in FIG. 1 . The above description of theDFA 1 of FIG. 1 is also valid for the DFA 1 according to FIG. 2 .

As shown in FIG. 2 the second part 2 of the DFA 1 comprises, besides theM DFTs 2 a, an overlap unit 2 b and M−1 delay units 2 c. The overlapunit 2 b is configured to generate in the time domain on the basis of acurrent block S a current overlap block s₁. The M−1 delay units 2 c areconfigured to generate in the time domain on the basis of the currentoverlap block s₁ M−1 delayed versions s₂, . . . , s_(M) of the currentoverlap block s₁. As described already above with respect to FIG. 1 ,the current overlap block s₁ comprises the samples of the current blockS and the N-L last consecutive samples of a directly previous block thatwas received by the DFA 1 directly before the current block S. The M−1delayed versions s₂, . . . , s_(M) are differently delayed versions ofthe current overlap block s₁. Each delay unit 2 c may be configured todelay the respective input by a multiple of Δ samples. During each clockthe DFA 1 may receive a new block.

Therefore, as shown in FIG. 2 the DFA 1 is configured to generate, onthe basis of the current block S, the current overlap block s₁ of thesize N greater than the size L and to generate the M−1 delayed versionss₂, . . . , s_(M) of the current overlap block s₁. The DFA 1 is furtherconfigured to jointly compute on the basis of the current overlap blocks₁ and the M−1 delayed versions s₂, . . . , s_(M) of the current overlapblock s₁ the M discrete Fourier transforms X₁, X₂, . . . , X_(M) byusing the M DFT filters 2 a. In other words, the DFA 1 is configured togenerate, by using the M DFT filters 2 a, the M discrete Fouriertransforms X₁, X₂, . . . , X_(M) on the basis of the current overlapblock s₁ and the M−1 delayed versions s₂, . . . , s_(M) of the currentoverlap block s₁. Each discrete Fourier transform X₁, X₂, . . . , X_(M)is of the size N and comprises N entries. For example, the discreteFourier transform X₁ comprises the N entries X_(1,1), X_(1,2), . . . ,X_(1, N); the discrete Fourier transform X₂ comprises the N entriesX_(2,1), X_(2,2), . . . , X_(2, N) and the discrete Fourier transformX_(M) comprises the N entries X_(M,1), X_(M,2), . . . , X_(M, N).

According to the embodiment of FIG. 2 , the DFA 1 is configured togenerate in the time domain the current overlap block s₁ on the basis ofthe current block S and the M−1 delayed versions s₂, . . . , s_(M) ofthe current overlap block s₁.

As shown in FIG. 2 , the DFA 1 may be configured to generate the M−1delayed versions s₂, . . . , s_(M) of the current overlap block s₁ byprogressively delaying the current overlap block s₁ in steps of Δsamples.

Further, as shown in FIG. 2 , the compensation filter 3 (second part) ofthe DFA 1 comprises N subfilters 3 ₁, 3 ₂, . . . , 3 _(N) (a subfiltermay also be referred to by only the term “filter”). Each subfiltercomprises a delay network 3 a and a linear combination algorithm 3 b.The k^(th) subfilter 3 _(k) (k=1, 2, . . . , N) of the compensationfilter 3 is configured to perform a filtering on the k^(th) entry of oneor more of the generated M discrete Fourier transforms X₁, X₂, . . . ,X_(M) to generate the k^(th) entry Y_(k) of the output Fourier transformY. In other words, the DFA 1 is configured to perform a filtering on thek^(th) entry of one or more of the generate M discrete Fouriertransforms X₁, X₂, . . . , X_(M) by using the k^(th) subfilter 3 _(k) togenerate the k^(th) entry Y_(k) of the output Fourier transform Y.

The description with respect to a k^(th) subfilter 3 _(k) of thecompensation filter 3 is valid for each subfilter of the N subfilters 3₁, 3 ₂, . . . , 3 _(N) of the compensation filter 3.

According to FIG. 2 , the k^(th) entries X_(1, k), X_(2, k), . . . ,X_(M, k) of the M discrete Fourier transforms X₁, X₂, . . . , X_(M) aresupplied to the k^(th) subfilter 3 _(k) of the compensation filter 3(k=1, 2, . . . , N). Nevertheless, for compensating the GVD by thecompensation filter 3, each subfilter 3 ₁, 3 ₂, . . . , 3 _(N) of thecompensation filter 3 does not need to perform a filtering on all k^(th)entries X_(1, k), X_(2, k), . . . , X_(M, k) of the M discrete Fouriertransforms X₁, X₂, . . . , X_(M). That is, each subfilter 3 ₁, 3 ₂, . .. , 3 _(N) is configured to perform a filtering on the k^(th) entry ofone or more discrete Fourier transforms of the M discrete Fouriertransforms X₁, X₂, . . . , X_(M). The subfilters 3 ₁, 3 ₂, . . . , 3_(N) may be different with respect to the one or more respective entriesof the one or more M discrete Fourier transforms X₁, X₂, . . . , X_(M)used for generating the respective entry Y₁, Y₂, . . . , Y_(N) of theoutput Fourier transform Y.

As shown in FIG. 2 , the k^(th) entries X_(1,k), X_(2,k), . . . ,X_(M,k) of the M discrete Fourier transforms X₁, X₂, . . . , X_(M) aresupplied to the delay network 3 a of the k^(th) subfilter. The delaynetwork 3 a of the k^(th) subfilter 3 _(k) is configured to generate thek^(th) entry of one or more delayed versions of one or more discreteFourier transforms (may be referred to as “one or more second discreteFourier transforms”) of the M discrete Fourier transforms X₁, X₂, . . ., X_(M) by delaying the k^(th) entry of the respective discrete Fouriertransform (second discrete Fourier transform) using one or more integerdelays. The delay network 3 a of the k^(th) subfilter may be configuredto generate the k^(th) entry of the one or more delayed versions of theone or more discrete Fourier transforms by progressively delaying thek^(th) entry of the respective discrete Fourier transform.

The one or more delayed versions of a discrete Fourier transform maydiffer to each other by a delay of multiples of L samples in the timedomain or a delay of multiples of blocks of the respective discreteFourier transform in the frequency domain. That is, a delayed version ofa discrete Fourier transform may correspond to the discrete Fouriertransform delayed by a delay of multiples of blocks of the discreteFourier transform in the frequency domain.

The k^(th) subfilter 3 _(k) is configured to generate, using therespective linear combination algorithm 3 b, the k^(th) entry of theoutput Fourier transform as a linear combination of the k^(th) entry ofone or more discrete Fourier transforms (may be referred to as “one ormore first discrete Fourier transforms”) of the M discrete Fouriertransforms X₁, X₂, . . . , X_(M) and/or the k^(th) entry of one or moredelayed versions of one or more discrete Fourier transforms (may bereferred to as “one or more second discrete Fourier transforms”) of theM discrete Fourier transforms X₁, X₂, . . . , X_(M). The k^(th) entry ofthe one or more delayed versions of the one or more second discreteFourier transforms is generated by the respective delay network 3 a ofthe k^(th) subfilter 3 k, as shown in FIG. 2 by the arrow between thebox “3a” indicating the delay network and the box “3b” indicating thelinear combination algorithm of each subfilter.

Some or all of the one or more first discrete Fourier transforms and ofthe one or more second discrete Fourier transforms may correspond toeach other.

The k^(th) subfilter may be configured to perform the linear combinationby weighting the k^(th) entry of the one or more first discrete Fouriertransforms, and/or the k^(th) entry of the one or more delayed versionsof the one or more second discrete Fourier transforms with a respectivecoefficient.

The k^(th) subfilter may be configured to linearly combine the k^(th)entry of the one or more first discrete Fourier transforms and/or thek^(th) entry of the one or more delayed versions of the one or morediscrete Fourier transforms by using one or more correspondingcoefficients of the linear combination algorithm.

As shown in FIG. 2 , the DFA 1 may optionally be configured to generate,on the basis of the output discrete Fourier transform Y, an output blocky of size N by using an inverse discrete Fourier transformation (IDFT)filter 4. Further, the DFA 1 may optionally be configured to generate anoutput block R of the size L on the basis of the output block y of thesize N, by using an overlap removal unit 4 a. The overlap removal unit 4a may generate the output block R of the size L on the basis of theoutput block y of the size N by using an overlap-add or an overlap-savemethod.

As described above, according to the embodiment of FIG. 2 the followingsteps may be performed for GVD compensation:

In a first step, each current block S of the size L (L=M·Δ) comprising Linput samples may be extended to a corresponding current overlap blocks₁ of size N (N=(M+m)·Δ) comprising N samples by using an overlaptechnique. The overlap technique may correspond to an overlap-add or anoverlap-save technique. The terms “overlap block” and “extended inputblock” may be used as synonyms.

In a second step, M−1 delayed versions s₂, . . . , s_(M) of therespective current overlap block s₁ may be generated. Each copy s₂, . .. , s_(M) of the respective current overlap block s₁ is successivelydelayed by Δ samples.

In a third step, M discrete Fourier transforms X₁, X₂, . . . , X_(M) maybe jointly computed, using the M DFT filters 2 a, on the basis of therespective current overlap block s₁ and the M−1 delayed versions of therespective current overlap block s₂, . . . , s_(M).

In a fourth step, the k^(th) entry Y_(k) of the output discrete Fouriertransform Y (k=1, . . . , N) may be computed, using the compensationfilter 3, as a linear combination of the k^(th) entry of one or moresuitable first discrete Fourier transforms of the M discrete Fouriertransforms X₁, X₂, . . . , X_(M) and/or the k^(th) entry of one or moresuitably delayed versions of one or more suitable second discreteFourier transforms of the M discrete Fourier transforms X₁, X₂, . . . ,X_(M). The sets of first and second discrete Fourier transforms mayoptionally have non-empty intersection. That is, optionally, some or allof the one or more first discrete Fourier transforms and of the one ormore second discrete Fourier transforms may correspond to each other.

In an optional fifth step, the output block y of size N may be computed,by using the IDFT filter 4, on the basis of the output discrete Fouriertransform Y.

In an optional sixth step, m·Δ overlapping samples may be removed fromthe output block y to obtain the block R of the size L (L=M·Δ)comprising L output samples. The optional overlap removal unit 4 a shownin FIG. 2 is configured to remove mA overlapping samples from the outputblock y to generate the block R of the size L. The samples of the blockR are GVD compensated with respect to the samples of the correspondingcurrent block S.

According to the embodiment of FIG. 2 the first step of overlapping andsecond step of delaying are performed in the time domain. According toanother embodiment the first step of overlapping and second step ofdelaying may also be performed in the frequency domain, as exemplarilydescribed below with respect to FIG. 6 in case M equals three (M=3), Lequals three samples (L=3) and N equals four samples (N=4).

Y_(k) [μ]=Σ_(i=0) ^(l−1)c_(k,i)·DFT_(k) (D^((i+P) ^(k) ^()·Δ)(Ŝ[μ])) Inthe following an implementation form of the above described fourth stepthat may be performed by an embodiment of the DFA 1 is described. Thek^(th) entry Y_(k)[μ](for k=1, . . . , N) of the N entries of the outputdiscrete Fourier transform Y may be computed as the following linearcombination

Y _(k)[μ]=Σ_(i=0) ^(l−1) c _(k,i) ·DFT _(k)(D ^((i+P) ^(k)^()·Δ)(Ŝ[μ])).  (1)

The k^(th) entry may also be referred to as the k^(th) discrete Fouriertransform (DFT) component (in short: k^(th) Fourier component) or as thek^(th) Fourier coefficient.

In the above equation (1), D^(n) represents a delay of n samples,DFT_(k) is a k^(th) Fourier coefficient (k^(th) entry) of a discreteFourier transform, Ŝ[μ] denotes the μ^(th) overlap block (μ^(th) block(input block) of the size L (L=M·Δ) extended by an overlap section ofthe size m·Δ), l is the number of coefficients in the linearcombination, c_(k,i) is a coefficient of the linear combination (whichmay be optimized as described below), and p_(k) is an integer number(which may be optimized as described below) corresponding to an integerdelay of p_(k) Δ samples. In particular, c_(k,i) may be a coefficient ofone or more coefficients of the linear combination algorithm 2 c of thek^(th) subfilter 3 _(k) of the compensation filter 3 (as shown below inequation (3)). Further, p_(k) may be an integer number corresponding toan integer delay of one or more integer delays of the delay network 2 bof the k^(th) subfilter 3 k (as shown below in equations (2) and (3)).

The term “D^((i+p) ^(k) ^()·Δ)(Ŝ[μ])”, inside the round brackets ofDFT_(k)( . . . ) in the above equation (1), corresponds to the input inthe time domain that is transformed by the respective DFT algorithm tothe frequency domain. Thus, DFT_(k) is the k^(th) entry of therespective discrete Fourier transform that is generated by the DFTalgorithm.

(i+p _(k))·Δ=q _(k,i) ·L+r _(k,i)∈{0, 1, . . . , M−1}p _(k)M Theequality

(i+p _(k))·Δ=q _(k,i) ·L+r _(k,i) ·Δq _(k,i) r _(k,i)∈{0,1, . . .,M−1}i+p _(k) M  (2)

(i+p_(k)) Δ=q_(k,i)·L+r_(k,i)·Δq_(k,i)r_(k,i) ∈{0, 1, . . . ,M−1}i+p_(k)Mholds, wherein the integers and denote the quotient and theremainder of the integer division of by.

Y_(k)[μ]=Σ_(i=1) ^(l−)1c_(k,i)·D^(q) ^(k,i) ^(·) ^(L) (DFT_(k)(D^(r)^(k,i) ^(·Δ)(Ŝ[μ])))=Σ_(i=1) ^(l−)1c_(k,i)·D^(q) ^(k,i) ^(·) ^(L)(X_((r) _(k,i) ₊1),k)k=1, . . . , Nq_(k,i)·Lq_(k,i)X_((r) _(k,i)_(+1),k)r_(k,i)r_(k,i)=0X_((r) _(k,i) _(+1),k)X_(1,k) Thus, the equation(1) may be rewritten as

Y _(k)[μ]=Σ_(i=1) ^(l−)1c _(k,i) ·D ^(q) ^(k,i) ^(·) ^(L) (DFT _(k)(D^(r) ^(k,i) ^(·Δ)(Ŝ[μ])))=Σ_(i=1) ^(l−)1c _(k,i) ·D ^(q) ^(k,i) ^(·)^(L) (X _((r) _(k,i) _(+1),k))k=1, . . . ,Nq _(k,i) ·Lq _(k,i) X _((r)_(k,i) _(+1),k) r _(k,i) r _(k,i)=0X _((r) _(k,i) _(+1),k) X_(1,k),  (3)

Y_(k)[μ]=Σ_(i=1) ^(l−)1c_(k,i)·D^(q) ^(k,i) ^(·) ^(L) (DFT_(k)(D^(r)^(k,i) ^(·Δ)(Ŝ[μ])))=Σ_(i=1) ^(l−)1c_(k,i)·D^(q) ^(k,i) ^(·) ^(L)(X_((r) _(k,i) ₊1),k)k=1, . . . , Nq_(k,i)·Lq_(k,i)X_((r) _(k,i+1),k)r_(k,i)r_(k,i)=0X_((r) _(k,i) _(+1),k)X_(1,k) wherein. In the aboveequation (3), a delay of a multiple of samples is implemented as delayof blocks of the corresponding discrete Fourier transform. In case theremainder is zero ( ), then the k^(th) entry shown in equation (3)corresponds to the k^(th) entry of the discrete Fourier transform X₁ ofthe M discrete Fourier transforms X₁, . . . , X_(M).

The above equation (3) shows that the linear combination according tothe above equation (1) is compatible with the block diagram of FIG. 2 .The above equation (3) may describe, according to an embodiment of theDFA 1, the filtering of the k^(th) entry X_((r) _(k,i+1),k) of adiscrete Fourier transform X_((r) _(k,i+1)) of the M discrete Fouriertransforms X₁, . . . , X_(M) by using the k^(th) subfilter 3 _(k). Foran embodiment of the DFA 1, the above equation (3) indicates for thek^(th) subfilter the delaying by the respective delay network 3 a (whichis represented by D^(q) ^(k,i) ^(·) ^(L) in the equation (3)) and thelinear combination by the respective linear combination algorithm 3 b(which is represented by c_(k,i) in the equation (3)).

In the following an optional optimization process of the coefficientsc_(k,i) and integer delays p_(k) for k=1, . . . , N and i=0, 1, . . . ,l−1 is described, in case the DFA 1 uses an overlap technique, such asan overlap-discard or overlap-add method, for generating a currentoverlap block of a current block received by the DFA 1.

For such an optimization the constraint is posed that the phase responseof the compensation filter 3 of the DFA 1 (which may also be referred toas equalizer filter) is the inverse of the GVD phase response, exceptfor an immaterial frequency-independent phase rotation and delay. Theoptimization target is the minimization of the time-domain aliasingresulting from the overlap technique (e.g. overlap-discard oroverlap-add method). In other terms, the target is to acquire an exactinversion of the GVD response in the frequency domain. Since theequivalent impulse response may exceed the overlap length (Δ·m), it is atarget that a proper measure (described below) of the resultingtime-domain aliasing is minimized.

Σ_(i=0) ^(l−)1c_(k,i)·exp(−j·2π·(i+p_(k))·Δ·f_(k))=h_(k) (k=1, . . . ,N)h_(k)−0.5≤f_(k)≤0.5 In mathematical terms the constraint may read

Σ_(i=0) ^(l−)1c _(k,i)·exp(−j·2π·(i+p _(k))·Δ·f _(k))=h _(k)(k=1, . . .,N)h _(k)−0.5≤f _(k)≤0.5,(4)

Σ_(i=0) ^(l−)1c_(k,i)·exp(−j·2π·(i+p_(k))·Δ·f_(k))=h_(k) (k=1, . . . ,N)h_(k)−0.5≤f_(k)≤0.5 wherein is the k^(th) Fourier coefficient (entry)of a discrete Fourier transform of the inverse GVD phase response and isthe normalized frequency associated with the k^(th) Fourier coefficient.

(a+1)X_(a)Y_(k) [μ]Z_(a,k) [μ]Z_(a,k) [μ]

Σ_(i=0) ^(l−)1c_(k,i)·DFT_(k) ((D^(r) ^(k,i) ^(·Δ)(Ŝ[μ]))·δ((i+p_(k))mod M, α)δ mod MM The aliasing generated by the compensation filter 3 inturn when only one of the M discrete Fourier transforms is non-zero isconsidered. When the -th Fourier transform is the only non-zerotransform of the M discrete Fourier transforms X₁, . . . , X_(M), itfollows from equation (1) that is equal to below:

(a+1)X _(a) Y _(k)[μ]Z _(a,k)[μ]Z _(a,k)[μ]

Σ_(i=0) ^(l−)1c _(k,i) ·DFT _(k)((D ^(r) ^(k,i) ^(·Δ)(Ŝ[μ]))·δ((i+p_(k))mod M,α)δ mod MM,  (5)

(a+1)X_(a)Y_(k) [μ]Z_(a,k) [μ]Z_(a,k) [μ]

Σ_(i=0) ^(l−)1c_(k,i)·DFT_(k) ((D^(r) ^(k,i) ^(·Δ)(Ŝ[μ]))·δ((i+p_(k))mod M, α)δ mod MM wherein a=0, . . . , M−1 and k=1, . . . , N; is theKronecker delta and denotes the reduction modulo.

DFT_(k)(a+1)DFT_(k) (D^((i+p) ^(k) ^()·Δ)(Ŝ[μ]))=1k=1, . . . ,N(a+1)a=0, 1, . . . , M−1 μz_(a)=IDFT([Σ_(i=0) ^(l−)1c_(1,i)·δ((i+p₁)mod M, a), . . . , Σ_(i=0) ^(l−)1c_(N,i)·δ((i+p_(N)) mod M,a)])z_(a)a=0, 1, . . . , M−1 In analogy with the case of theconventional impulse response, for the purpose of quantifying thetime-domain aliasing, it is assumed that all Fourier coefficients of the-th discrete Fourier transform according to equation (5) are equal toone (for). The resulting -th time-domain response for is considered inthe following equation (6), where the time index is dropped:

DFT _(k)(a+1)DFT _(k)(D ^((i+p) ^(k) ^()·Δ)(Ŝ[μ]))=1k=1, . . .,N(a+1)a=0,1, . . . ,M−1μz _(a)=IDFT([Σ_(i=0) ^(l−)1c _(1,i)·δ((i+p₁)mod M,a), . . . ,Σ_(i=0) ^(l−)1c _(N,i)·δ((i+p _(N))mod M,a)])z _(a)a=0,1, . . . ,M−1.  (6)

DFT_(k)(a+1)DFT_(k) (D^((i+p) ^(k) ^()·Δ)(Ŝ[μ]))=1k=1, . . . ,N(a+1)a=0, 1, . . . , M−1 μz_(a)=IDFT([Σ_(i=0) ^(l−)1c_(1,i)·δ((i+p₁)mod M, a), . . . , Σ_(i=0) ^(l−)1c_(N,i)·δ((i+p_(N)) mod M,a)])z_(a)a=0, 1, . . . , M−1A suitable norm of the subsequence of 0 thatlies outside of the overlap region is regarded as time-domain aliasing.Therefore, the target of the optimization problem is the minimization ofthis norm.

Several choices of the norm are possible. If the infinite norm or the L₁distance (L₁ norm) are used, a mixed-integer linear programming problemis obtained. In case of the Euclidean norm, a least-square problem isobtained. Both types of problems may be solved by using standardnumerical routines.

The solution of the optional optimization problem may be done by a usualoptimization routine. The approach to define the optional optimizationproblem consists of two steps:

In a first step the response of the DFA 1 between the points after theoverlap unit 2 b and before the overlap removal unit 4 a is constrainedto be the inverse of the GVD function. In a second step the time-domainaliasing is minimized. The time-domain aliasing is defined using aproper norm of M equivalent impulse responses z_(a) for a=0, 1, . . . ,M−1. The (a+1)-th equivalent impulse response is obtained when all theFourier coefficients of the (a+1)-th Fourier transform X_(a) of the Mdiscrete Fourier transforms X₁, . . . , X_(M) are equal to 1 and all theFourier coefficients of the remaining M−1 Fourier transforms X_(i) (i∈{1, 2, . . . , M}, i≠a+1) of the M discrete Fourier transforms X₁, . .. , X_(M) are equal to 0.

A DFA that is equivalent to the DFA of FIG. 2 may optionally beimplemented with a single DFT filter and M IDFT filters according to thetheory of signal flow graphs (SFGs). This implementation may be referredto as the transposed implementation. The transposed implementationallows the same overlap reduction (reduction of the overlap ratio) asthe implementation of the DFA 1 shown in FIG. 2 . However, thetransposed implementation does not allow the joint IDFT computation andis therefore less computationally efficient.

FIG. 3 shows an example of a transformation from the time domain to thefrequency domain and a delaying in the frequency domain used in adigital filter arrangement (DFA) according to an embodiment of thepresent disclosure.

According to FIG. 3 a current block S(μ) of size L (L=M·Δ) may be splitin the time domain into M subblocks respectively sequences s(μ·M),s(μ·M+1), . . . , s((μ+1)M−1) of size Δ(M and Δ are positive integers),wherein “μ” corresponds to a time point, such as the time point of theclock of a processor for executing the function of the DFA. Thus, incase μequals zero (μ=0) and M equals three (M=3), the current block S(0)may be split into the three consecutive subblocks s(0), s(1) and s(2).On the basis of each subblock corresponding M discrete Fouriertransforms DFT([0, 0, . . . , 0, s(μ·M)]), DFT([0, 0, . . . , 0,s(μ·M+1)]) and DFT([0, 0, . . . , 0, s((μ+1)·M−1)](which may also bereferred to as “M further discrete Fourier transforms” or “Minput-pruned discrete Fourier transforms”) are generated. The generationof these M further discrete Fourier transforms according to anembodiment of the present disclosure is described below with respect toFIG. 5 .

Further, as shown in FIG. 3 one or more delayed versions of the Mfurther discrete Fourier transforms may be generated in the frequencydomain. For example, in case μequals zero (μ=0) and M equals 3 (M=3),for the three subblocks s(0), s(1) and s(2) the three further discreteFourier transforms DFT([0, 0, . . . , 0, s(0)]), DFT([0, 0, . . . , 0,s(1)]) and DFT([0, 0, . . . , 0, s(2)]) may be generated. As shown inFIG. 3 , for these three further discrete Fourier transforms DFT([0, 0,. . . , 0, s(0)]), DFT([0, 0, . . . , 0, s(1)]) and DFT([0, 0, . . . ,0, s(2)]) the delayed versions DFT([0, 0, . . . , 0, s(−3)]), DFT([0, 0,. . . , 0, s(−2)]) and DFT([0, 0, . . . , 0, s(−1)]) may be generated inthe frequency domain. The three subblocks s(−3), s(−2) and s(−1)correspond to the three consecutive subblocks of the block S(−1), incase M equals three (M=3). The block S(−1) is the directly previousblock to the block S(0) in the sequence of consecutive blocks that maybe input to the digital filter arrangement of the first aspect or any ofits implementation forms for inputting a sequence of samples of adigital input signal in the time domain. The subblock s(−3) of the blockS(−1) corresponds to the subblock s(0) of the block S(0), the subblocks(−2) of the block S(−1) corresponds to the subblock s(1) of the blockS(0) and the subblock s(−1) of the block S(−1) corresponds to thesubblock s(2) of the block S(0). Thus, the delayed version DFT([0, 0, .. . , 0, s(−3)]) corresponds to the further discrete Fourier transformof the subblock s(−3), the delayed version DFT([0, 0, . . . , 0, s(−2)])corresponds to the further discrete Fourier transform of the subblocks(−2) and the delayed version DFT([0, 0, . . . , 0, s(−1)]) correspondsto the further discrete Fourier transform of the subblock s(−1).

FIG. 4 shows an example of an aligning and adding of discrete Fouriertransforms in the frequency domain.

According to FIG. 4 an alignment and adding of discrete Fouriertransforms in the frequency domain may be performed by a symbol-wisemultiplication of the discrete Fourier transforms by a rotation vectorρ_(M+m) whose elements are powers of the elements of a base rotationvector ρ_1_(M+m) and a subsequent addition of the results of thesymbol-wise multiplications.

${{\rho\_}1_{M + m}{\rho\_}1_{M + m}}\overset{\bigtriangleup}{=}{{\exp\left( {j{\frac{2\pi}{M + m}\left\lbrack {0,1,\ldots,{N - 1}} \right\rbrack}} \right)}\left( {{\rho\_}1_{M + m}} \right)^{M - 1}}$

The base rotation vector may be as follows:

${{{\rho\_}1_{M + m}{\rho\_}1_{M + m}}\overset{\bigtriangleup}{=}{{\exp\left( {j{\frac{2\pi}{M + m}\left\lbrack {0,1,\ldots,{N - 1}} \right\rbrack}} \right)}\left( {{\rho\_}1_{M + m}} \right)^{M - 1}}},$

wherein j is the imaginary unit.

${{\rho\_}1_{M + m}{\rho\_}1_{M + m}}\overset{\bigtriangleup}{=}{{\exp\left( {j{\frac{2\pi}{M + m}\left\lbrack {0,1,\ldots,{N - 1}} \right\rbrack}} \right)}\left( {{\rho\_}1_{M + m}} \right)^{M - 1}}$

For example, as can be seen from FIG. 4 , by performing the symbol-wisemultiplication of the discrete Fourier transform DFT([0, 0, . . . , 0,s₀]) with an alignment may be performed. Namely, the samples of theblock so in the sequence of [0, 0, . . . , 0, s₀] that is the input ofthe discrete Fourier transform DFT([0, 0, . . . , 0, s₀]) is shiftedfrom the one end (right side) of the sequence to the other end (leftside) of the sequence providing the sequence [s₀, 0, . . . , 0, 0] thatis the input of the discrete transform DFT([s_(0, 0), . . . , 0, 0]).This symbol-wise multiplication by a rotation vector corresponds to ashift in the time domain. As shown in FIG. 4 , by aligning and addingthe M discrete Fourier transforms DFT([0, 0, . . . , 0, s₀]), DFT([0, 0,. . . , 0, s₁]), . . . , DFT([0, 0, . . . , 0, s_(M−1)]) the discreteFourier transform DFT([s₀, s₁, . . . , s_(M−1)]) may be generated in thefrequency domain.

FIG. 5 shows the generation of the M further discrete Fourier transformsshown in FIGS. 3 and 6 by a digital filter arrangement (DFA) accordingto an embodiment of the present disclosure.

The above description of the DFA according to the first aspect and itsimplementation forms is correspondingly valid for the DFA of FIG. 5 .

According to an embodiment of the present disclosure, the generation ofthe M further discrete Fourier transforms (which may also be referred toas “input-pruned discrete Fourier transforms”) shown in FIGS. 3 and 6may comprise the following steps:

The DFA is configured to group the Δ samples of each subblock of the Msubblocks into two parts of samples x− and x+. Optionally, the DFA maybe configured to group the A samples of each subblock of the M subblocksinto two equal parts of samples x− and x+. The M subblocks form acurrent block S of the size L (L=M·Δ) received by the DFA (not shown inFIG. 5 ).

In a second step S51 following the first step, the DFA is configured tocarry out a zero-padding of each subblock to a zero-padded sequence ofthe size Γ (Γ>Δ) by adding Γ-A zeros between the two parts of samples x−and x+ (Γ is smaller than N). Γ may beneficially be a positive integer.As a result in the second step S51 starting with a subblockcorresponding to the sequence of samples [x−, x+] the zero-paddedsequence [x+, 0, . . . , 0, x−] is generated, wherein the zero-paddedsequence [x+, 0, . . . , 0, x−] comprises the two parts of samples x−and x+ of the subblock and Γ-Δ zeros. The zero-padding introduces excessbandwidth and allows for interpolation, which is performed in the stepS53. According to an embodiment, Γ may be assumed to be

$\frac{N}{3}{\left( {\Gamma = \frac{N}{3}} \right).}$

As shown in FIG. 5 , the DFA may be configured to group the Δ samples ofeach subblock into a first part x+ of samples (first part comprisingconsecutive coefficients) and a second part x− of samples (second partcomprising consecutive coefficients). The number of samples of the firstand second part may optionally be the same. The term “contiguous” may beused as a synonym for the term “consecutive”.

The first and second part x+ and x− may be arranged in the respectivesubblock, such that in the sequence of consecutive samples of therespective subblock the samples of the second part x− are arrangedprevious to the samples of the first part x+. That is, the subblock maycorrespond to the sequence [x−, x+]. That is, the samples of the secondpart x− may be arranged at the beginning and the samples of the firstpart x+ may be arranged at the end of the respective subblock.

As shown in FIG. 5 , the zero-padding in the second step S51 of eachsubblock generates a zero-padded sequence [x+, 0, . . . , 0, x−] of thesize Γ by adding Γ-Δ zeros between the first part x+ of samples and thesecond part x− of samples of the respective subblock, wherein thepositions of the first part x+ and second part x− are swapped (thesequence [x−, x+] is zero-padded to the sequence [x+, 0, . . . , 0,x−]). Thus, after zero-padding, the samples of the first part x+ may bearranged at the beginning of the zero-padded sequence and the samples ofthe second part x− may be arranged at the end of the zero-paddedsequence, wherein the Γ-Δ zeros are arranged between these two parts.The zero-padding of each subblock may entail a reordering of the twoparts of samples of the respective subblock.

In a third step S52 following the second step S51, the zero-paddedsequence generated in the second step S51 is transformed by a DFTalgorithm of the size Γ to a discrete Fourier transform DFT_(Γ)([x+, 0,. . . , 0, x−]) of the size F. The DFT algorithm of the size Γ may be afast Fourier transform (FFT) algorithm.

In a fourth step S53 following the second step S52, the DFA isconfigured to interpolate, using an interpolation algorithm, the Γsamples of the discrete Fourier transform DFT_(Γ)([x+, 0, . . . , 0,x−]) of the size F, generated in the step S52, to N samples of anotherdiscrete Fourier transform DFT_(N)([x+, 0, . . . , 0, x−]) of the size N(N=(M+m)·Δ). The discrete Fourier transform DFT_(Γ)([x+, 0, . . . , 0,x−]) of the size Γ may be interpolated from Γ samples to N samples usinga low-pass real filter.

In a fifth step S54 following the fourth step S53, the DFA is configuredto perform a symbol-wise multiplication of the samples of the otherdiscrete Fourier transform DFT_(N)([x+, 0, . . . , 0, x−]) by a rotationvector ρ_2_(M+m) to obtain the respective further Fourier transformDFT_(N)([0, . . . , 0, x−, x+,]) of the size N.

The symbol-wise multiplication of the samples of each other discreteFourier transform by the rotation vector in step S54 corresponds to ashift in the time domain of the non-zero samples of the respectivezero-padded sequence. Therefore, as shown in FIG. 5 , the first andsecond part x+ and x− are arranged together at the end of the sequence[0, . . . , 0, x−, x+] to which the further discrete Fourier transformDFT_(N)([0, . . . , 0, x−, x+,]) corresponds to, wherein the samples ofthe second part x− are arranged previous to the samples of the firstpart x+.

${{\rho\_}2_{M + m}{\rho\_}2_{M + m}}\overset{\bigtriangleup}{=}{\exp\left( {j{\frac{\pi}{M + m}\left\lbrack {0,1,\ldots,{N - 1}} \right\rbrack}} \right)}$

The rotation vector may be as follows:

${{{\rho\_}2_{M + m}{\rho\_}2_{M + m}}\overset{\bigtriangleup}{=}{\exp\left( {j{\frac{\pi}{M + m}\left\lbrack {0,1,\ldots,{N - 1}} \right\rbrack}} \right)}},$

wherein j is the imaginary unit.

The interpolation may be implemented by using a polyphase finite impulseresponse (FIR) filter, wherein the input of the polyphase FIR filter isregarded as periodic.

The DFT algorithm of the above described third step S52 and theinterpolation algorithm of the above described fourth step S53 implementa DFT filter of the M DFT filters used by the DFA for generating the Mfurther discrete Fourier transforms. As shown with respect to FIG. 6 ,on the basis of the M further discrete Fourier transforms the M discreteFourier transforms, whose entries are provided to the compensationfilter of the DFA, may be generated. In particular, the DFA may beconfigured to generate the M discrete Fourier transforms by delaying theM further discrete Fourier transforms and by aligning and adding one ormore of the M delayed further Fourier transforms and one or more of theM further discrete Fourier transforms.

FIG. 6 shows a generation of M discrete Fourier transforms of a currentoverlap block of a size N greater than the size L and of M−1 delayedversions of the current overlap block by using M discrete Fouriertransform (DFT) filters of a digital filter arrangement (DFA) accordingto an embodiment of the present disclosure, wherein M equals three(M=3), L equals three times Δ samples (L=3·Δ), Δ is a generic positiveinteger number and N equals four times Δ samples (N=4·Δ). The overlapratio of the DFA of FIG. 6 corresponds to one-fourth

$\left( {\frac{m}{\left( {M + m} \right)} = \frac{1}{4}} \right).$

The above description of the DFA according to the first aspect and itsimplementation forms is correspondingly valid for the DFA of FIG. 6 .

The DFA of FIG. 6 corresponds to the DFA 1 of FIG. 1 , wherein FIG. 6shows an implementation form of the first part 2 of the DFA. The abovedescription of the DFA 1 of FIG. 1 is also valid for the DFA accordingto FIG. 6 .

As shown in FIG. 6 , the DFA, in particular the first part 2 of the DFA,is configured to perform the following steps for generating the Mdiscrete Fourier transforms X₁, . . . , X_(M):

In a first step S61 the DFA is configured to split a current block Srespectively S(μ) into M subblocks each of the size Δ, wherein eachsubblock comprises Δ samples of the current block S. A subblock of thesize Δ corresponds to a sequence of samples of the length A. Forexample, as shown in FIG. 6 , M may be equal to three (M=3). Thus, thecurrent block S corresponds to a sequence of 3 Δ samples [z_(k-2),z_(k-1), z_(k)], wherein z_(k) is the subblock of Δ samples at the endof the sequence of the 3·Δ samples of the current block S and z_(k-2) isthe subblock of A samples at the beginning of the sequence of the 3·Δsamples of the current block S. Thus, in the sequence of blocks receivedby the DFA, the subblock of samples z_(k-2) corresponds to the samplesof the current block S that follow directly after the last subblock ofsamples z_(k-3) of the directly previous block that was received by theDFA directly before the current block S. The directly previous blockcorresponds to the sequence of the 3·Δ samples [z_(k-5), z_(k-4),z_(k-3)](in case M=3).

As shown in FIG. 6 , in the first step S61 the DFA is configured tosplit the current block S corresponding to the sequence [z_(k-2),z_(k-1), z_(k)](in case M=3) into M subblocks z_(k), z_(k-1), andz_(k-2) each of the size Δ, wherein each subblock comprises Δ samples ofthe current block S.

In a second step S62 following the first step S61, the DFA is configuredto zero-pad each subblock to a zero-padded sequence of the size Γcomprising Γ samples (Γ<N), wherein each zero-padded sequence comprisesΔ samples of the corresponding subblock and Γ-Δ zeros. Further, in thesecond step S62 the DFA is configured to jointly approximate on thebasis of the M zero-padded sequences, M further discrete Fouriertransforms (input-pruned Fourier transforms) of the size N (N=Δ. (M+m))by using the M DFT filters (each DFT filter is implemented by a DFTalgorithm of the size Γ and an interpolation algorithm). For example mmay equal to one (m=1). In this case, as shown in FIG. 6 , starting fromthe subblock z_(k) the further discrete Fourier transform DFT([0, 0, 0,z_(k)]), starting from the subblock z_(k-1), the further discreteFourier transform DFT([0, 0, 0, z_(k-1)]) and from the subblock z_(k-2)the further discrete Fourier transform DFT([0, 0, 0, z_(k-2)]) may begenerated in the second step S62. For a detailed description for theimplementation of the second step S62 of the DFA according to anembodiment of the present disclosure reference is made to thedescription of FIG. 5 .

The DFA is configured to generate the M (e.g. M=3) discrete Fouriertransforms X₁, X₂, X₃ on the basis of the M further discrete Fouriertransforms DFT([0, 0, 0, z_(k)]), DFT([0, 0, 0, z_(k-1)]) and DFT([0, 0,0, z_(k-2)]), generated in the second step S62, by performing a thirdstep S63 following the second step S62 and a fourth step S64 followingthe third step S63. In the third step S63, the DFA is configured todelay the M further discrete Fourier transforms to generate M delayedfurther discrete Fourier transforms DFT([0, 0, 0, z_(k-3)]), DFT([0, 0,0, z_(k)-4]), DFT([0, 0, 0, z_(k)5]). In the third step the M furtherdiscrete Fourier transforms are delayed in order to provide with thedelayed versions enough samples of a directly previous block received bythe DFA directly before the current block S or of more than one previousblock received by the DFA before the current block S for generating theM discrete Fourier transforms X₁, X₂ and X₃.

In the fourth step S64, the DFA is configured to align and add one ormore of the M delayed further discrete Fourier transforms DFT([0, 0, 0,z_(k)3]), DFT([0, 0, 0, z_(k)4]), DFT([0, 0, 0, z_(k-5)]) and one ormore of the M further discrete Fourier transforms DFT([0, 0, 0, z_(k)]),DFT([0, 0, 0, z_(k-1)], DFT([0, 0, 0, z_(k-2)]) to generate the Mdiscrete Fourier transforms X₁, X₂, X₃. The time shift necessary toalign the one or more further discrete Fourier transforms and one ormore delayed further discrete Fourier transforms is implemented in thefrequency domain via a symbol-wise multiplication of the respectivefurther discrete Fourier transforms and respective delayed furtherdiscrete Fourier transforms by suitable powers of the rotation vectorρ_1_(M+m), (which corresponds to p_1₄ in case M=3 and m=1), as shown inthe FIG. 6 . The rotation vector may be as follows:

${{{\rho\_}1_{M + m}}\overset{\bigtriangleup}{=}{\exp\left( {j{\frac{2\pi}{M + m}\left\lbrack {0,1,\ldots,{N - 1}} \right\rbrack}} \right)}},$

where j is the imaginary unit. For a further description of the fourthstep S64 of aligning and adding reference is made to the description ofFIG. 4 .

As shown in FIG. 6 , a first discrete Fourier transform X₁ of the Mdiscrete Fourier transforms X₁, X₂, X₃ (M=3) corresponds to the discreteFourier transform of the sequence [z_(k-3), z_(k-2), z_(k-1), z_(k)].Thus, the first discrete Fourier transform X₁ corresponds to thediscrete Fourier transform of a current overlap block of the currentblock S, wherein the current block S corresponds to the sequence[z_(k-2), z_(k-1), z_(k)] in the time domain and the correspondingoverlap block corresponds to the sequence [z_(k-3), z_(k-2), z_(k-1),z_(k)] in the time domain. Namely, as outlined above, the currentoverlap block comprises the samples of the corresponding current blockand the N-L last consecutive samples of a directly previous block S thatwas received by the DFA directly before the respective current block S.Since in FIG. 6 it is assumed that M corresponds to three and mcorresponds to one (M=3, m=1), N corresponds to 4·Δ samples and Lcorresponds to 3·Δ samples (N=(M+m) A=4·Δ and L=ΔM=3·Δ). Thus, accordingto FIG. 6 , the current overlap block comprises the 3·Δ samples of thecorresponding current block S and the last consecutive Δ samples(N−L=4·Δ−3·Δ=Δ) of a directly previous block S that was received by theDFA directly before the respective current block S. As outlined above,the last Δ samples of the directly previous block S corresponds to thesamples z_(k-3).

Further, as shown in FIG. 6 , a second discrete Fourier transform X₂ ofthe M discrete Fourier transforms X₁, X₂, X₃ corresponds to the discreteFourier transform of the sequence [z_(k-4), z_(k-3), z_(k-2), z_(k-1)].Thus, the second discrete Fourier transform X₂ corresponds to thediscrete Fourier transform of a delayed version of the current overlapblock S. In particular, this delayed version corresponds to the currentoverlap block delayed by Δ samples. Furthermore, as shown in FIG. 6 , athird discrete Fourier transform X₃ of the M discrete Fourier transformsX₁, X₂, X₃ corresponds to the discrete Fourier transform of the sequence[z_(k-5), z_(k-4), z_(k-3), z_(k-2)]. Thus, the third discrete Fouriertransform X₃ corresponds to the discrete Fourier transform of a delayedversion of the current overlap block S. In particular, this delayedversion corresponds to the current overlap block delayed by 2Δ samples.

The above description is also valid in case M, m, and N have differentvalues. FIG. 6 shows by way of example the case in which M=3 and m=1.

Therefore, the DFA according to FIG. 6 is configured to generate Mdiscrete Fourier transforms X₁, X₂, X₃ (M=3) of a current overlap block[z_(k-3), z_(k-2), z_(k-1), z_(k)] and of M−1 delayed versions(M−1=3−1=2) of the current overlap block [Z_(k-4), z_(k-3), z_(k-2),z_(k-1)] and [z_(k-5), z_(k-4), z_(k-3), z_(k-2)].

Thus, the DFA of FIG. 6 is configured to provide the M discrete Fouriertransforms X₁, . . . , X_(M) for a filtering by the compensation filter(shown in FIG. 2 , but not shown in FIG. 6 ) of the DFA. According tothe embodiment of FIG. 2 the step of overlapping and step of delayingare performed in the time domain. In contrast thereto, according to theembodiment of FIG. 6 the first step of overlapping and step of delayingis performed in the frequency domain. The filtering of the M discreteFourier transforms X₁, . . . , X_(M) (X₁, . . . , X₃ in case M=3) by thecompensation filter of the DFA may be performed as described above withrespect to FIGS. 1 and 2 .

FIG. 7A shows a transmitter according to an embodiment of the presentdisclosure.

The transmitter 5 of FIG. 7A is configured to transmit, on the basis ofa digital signal, a modulated light signal via one or more opticalfibers to a receiver (not shown in FIG. 7A). The transmitter 5 comprisesa digital filter arrangement (DFA) 1 according to the first aspect orany of its implementation forms and is configured to pre-compensategroup velocity dispersion (GVD) of the modulated light signal on thebasis of the digital signal using the DFA 1. The above description withrespect to the DFA according to the first aspect or any of itsimplementation forms and the above description of the FIGS. 1 to 6 isvalid for the DFA 1 of the transmitter 5.

The above description with respect to the transmitter of the secondaspect or any of its implementation forms is also valid for thetransmitter 5 of FIG. 7A.

FIG. 7B shows a receiver according to an embodiment of the presentdisclosure.

The receiver 6 of FIG. 7B is configured to receive a modulated lightsignal via one or more optical fibers from a transmitter and to convertthe received modulated light signal into a digital signal (not shown inFIG. 7B). The receiver 6 comprises a digital filter arrangement (DFA) 1according to the first aspect or any of its implementation forms and isconfigured to compensate group velocity dispersion (GVD) of the receivedmodulated light signal on the basis of the digital signal using the DFA1. The above description with respect to the DFA according to the firstaspect or any of its implementation forms and the above description ofthe FIGS. 1 to 6 is valid for the DFA 1 of the receiver 6.

The above description with respect to the receiver of the third aspector any of its implementation forms is also valid for the receiver 6 ofFIG. 7B.

FIG. 8 shows an optical transmission system (OTS) according to anembodiment of the present disclosure.

The OTS 7 of FIG. 8 comprises one or more optical fibers 10, atransmitter 8 and a receiver 9. The transmitter 8 is configured totransmit, on the basis of a first digital signal, a modulated lightsignal via the one or more optical fibers 10 to the receiver 9. Thereceiver 9 is configured to receive the modulated light signal via theone or more optical fibers 10 from the transmitter 8 and convert thereceived modulated light signal into a second digital signal.

The transmitter 8 may be a transmitter according to the second aspect orany of its implementation forms (such as the transmitter 5 of FIG. 7A),which is configured to pre-compensate group velocity dispersion (GVD) ofthe modulated light signal on the basis of the first digital signalusing the digital filter arrangement (DFA) according to the first aspector any of its implementation forms.

Alternatively or additionally, the receiver 9 may be a receiveraccording to the third aspect or any of its implementation forms (suchas the receiver 6 of FIG. 7B), which is configured to compensate GVD ofthe received modulated light signal on the basis of the second digitalsignal using the DFA according to the first aspect or any of itsimplementation forms.

The above description with respect to the OTS of the fourth aspect orany of its implementation forms is also valid for the OTS 7 of FIG. 8 .

The present disclosure has been described in conjunction with variousembodiments as examples as well as implementations. However, othervariations can be understood and effected by those persons skilled inthe art and practicing the claimed invention, from the studies of thedrawings, this disclosure and the independent claims. In the claims aswell as in the description the word “comprising” does not exclude otherelements or steps and the indefinite article “a” or “an” does notexclude a plurality. A single element or other unit may fulfill thefunctions of several entities or items recited in the claims. The merefact that certain measures are recited in the mutual different dependentclaims does not indicate that a combination of these measures cannot beused in an advantageous implementation.

What is claimed is:
 1. A digital filter arrangement (DFA), comprising: areceiver; M discrete Fourier transform (DFT) filters, wherein each DFTfilter of the M DFT filters is implemented by a fast Fourier transform(FFT) algorithm of a size Γ smaller than a size N and by aninterpolation algorithm; and a compensation filter, implemented by adelay network and a linear combination algorithm; wherein the receiveris configured to: receive a sequence of samples of a digital inputsignal in the time domain in the form of consecutive blocks of size L,wherein each block of the blocks comprises L consecutive samples of thedigital input signal; wherein the M DFT filters are configured to:generate M discrete Fourier transforms of a current overlap block and ofM−1 delayed versions of the current overlap block, wherein the currentoverlap block is of the size N and the size N is greater than the sizeL, wherein each generated discrete Fourier transform is of the size Nand comprises N entries, the current overlap block comprises samples ofa current block of the blocks and the N-L last consecutive samples of adirectly previous block that was received by the DFA directly before thecurrent block of the blocks, and wherein the compensation filter isconfigured to: filter the N entries of the generated M discrete Fouriertransforms to generate an output discrete Fourier transform with Nentries.
 2. The DFA according to claim 1, further comprising: an inversediscrete Fourier transformation (IDFT) filter, configured to generate,on the basis of the output discrete Fourier transform, an output blockof size N.
 3. The DFA according to claim 2, wherein the DFA isconfigured to: generate an output block of the size L on the basis ofthe output block of the size N, by using an overlap-add or anoverlap-save method.
 4. The DFA according to claim 1, wherein the size Lof each block is the product of M and Δ, L=M·Δ, wherein M and Δ are eacha positive integer.
 5. The DFA according to claim 1, wherein: the size Nof the current overlap block is the product of Δ with the sum of M andm, N=Δ·(M+m), wherein m is a positive integer, and the product of Δ andm, A m, corresponds to the number of last consecutive samples in thesequence of samples of the directly previous block.
 6. The DFA accordingto claim 1, wherein the DFA is configured to generate, on the basis ofthe current block, the current overlap block of the size N and greaterthan the size L; and generate the M−1 delayed versions of the currentoverlap block; and wherein the M DFT filters are configured to: jointlyapproximate, on the basis of the current overlap block and the M−1delayed versions of the current overlap block, the M discrete Fouriertransforms.
 7. The DFA according to claim 6, wherein the DFA isconfigured to generate the M−1 delayed versions of the current overlapblock by progressively delaying the current overlap block in steps of Δsamples.
 8. The DFA according to claim 1, wherein the DFA is configuredto split the current block into M subblocks each of a size Δ, whereineach subblock comprises Δ consecutive samples of the current block; andzero-pad each subblock to a zero-padded sequence of the size Γcomprising F samples, wherein each zero-padded sequence comprises the Δsamples of the corresponding subblock and Γ-A zeros; and wherein the MDFT filters are configured to: jointly approximate, on the basis of theM zero-padded sequences, M further discrete Fourier transforms; andgenerate the M discrete Fourier transforms by: delaying the M furtherdiscrete Fourier transforms, and aligning and adding one or more of theM delayed further discrete Fourier transforms and one or more of the Mfurther discrete Fourier transforms.
 9. The DFA according to claim 8,wherein the M DFT filters are configured to perform the alignment byperforming a symbol-wise multiplication of one or more delayed furtherdiscrete Fourier transforms or one or more further discrete Fouriertransforms by a rotation vector.
 10. The DFA according to claim 8,wherein the DFA is configured to: group the Δ samples of each subblockinto two parts of samples; carry out the zero-padding of each subblockto a zero-padded sequence of the size Γ by adding Γ-Δ zeros between thetwo parts of samples; and wherein the M DFT filters are configured tojointly approximate, on the basis of the M zero-padded sequences, the Mfurther discrete Fourier transforms.
 11. The DFA according to claim 10,wherein each of the M DFT filters are configured to jointly approximate,on the basis of the M zero-padded sequences, the M further discreteFourier transforms by: transforming each zero-padded sequence by the FFTalgorithm of the size Γ to a discrete Fourier transform of the size F;interpolating the Γ samples of each discrete Fourier transform of thesize Γ to N samples of another discrete Fourier transform of the size Nby a low-pass filter; and performing a symbol-wise multiplication of thesamples of each other discrete Fourier transform by a rotation vector toobtain the respective further discrete Fourier transform.
 12. The DFAaccording to claim 1, wherein the compensation filter comprises Nsubfilters, wherein each subfilter is implemented by the delay networkand the linear combination algorithm; and wherein the DFA is configuredto perform a filtering on a k^(th) entry of one or more of the generatedM discrete Fourier transforms by using a k^(th) subfilter to generate ak^(th) entry of the output Fourier transform, wherein each discreteFourier transform comprises N entries.
 13. The DFA according to claim 1,wherein the compensation filter comprises N subfilters, and a k^(th)subfilter of the N subfilters is configured to generate a k^(th) entryof the output Fourier transform as a linear combination of: a k^(th)entry of one or more first discrete Fourier transforms of the M discreteFourier transforms; or a k^(th) entry of one or more delayed versions ofone or more second discrete Fourier transforms of the M discrete Fouriertransforms.
 14. The DFA according to claim 13, wherein the k^(th)subfilter is configured to perform the linear combination by weightingthe k^(th) entry of the one or more first discrete Fourier transforms,or the k^(th) entry of the one or more delayed versions of the one ormore second discrete Fourier transforms with a respective coefficient.15. The DFA according to claim 13, wherein the k^(th) subfilter isconfigured to generate the k^(th) entry of each of the one or moredelayed versions of each of the one or more second discrete Fouriertransforms by delaying the k^(th) entry of the respective seconddiscrete Fourier transform using one or more integer delays.
 16. Anoptical transmission system (OTS), comprising: one or more opticalfibers; a transmitter; and a receiver; wherein the transmitter isconfigured to transmit, on the basis of a first digital signal, amodulated light signal via the one or more optical fibers to thereceiver; wherein the receiver is configured to receive the modulatedlight signal via the one or more optical fibers from the transmitter andconvert the received modulated light signal into a second digitalsignal; wherein: the transmitter is configured to pre-compensate groupvelocity dispersion (GVD) of the modulated light signal on the basis ofthe first digital signal using a digital filter arrangement (DFA), orthe receiver is configured to compensate GVD of the received modulatedlight signal on the basis of the second digital signal using a DFA; andwherein the DFA comprises: a DFA receiver; M discrete Fourier transform(DFT) filters, wherein each DFT filter of the M DFT filters isimplemented by a fast Fourier transform (FFT) algorithm of a size Γsmaller than a size N and by an interpolation algorithm; and acompensation filter, implemented by a delay network and a linearcombination algorithm; wherein the DFA receiver is configured to:receive a sequence of samples of a digital input signal in the timedomain in the form of consecutive blocks of size L, wherein each blockof the blocks comprises L consecutive samples of the digital inputsignal; wherein the M DFT filters are configured to: generate M discreteFourier transforms of a current overlap block and of M−1 delayedversions of the current overlap block, wherein the current overlap blockis of the size N and the size N is greater than the size L, wherein eachgenerated discrete Fourier transform is of the size N and comprises Nentries, the current overlap block comprises samples of a current blockof the blocks and the N-L last consecutive samples of a directlyprevious block that was received by the DFA directly before the currentblock of the blocks, and wherein the compensation filter is configuredto: filter the N entries of the generated M discrete Fourier transformsto generate an output discrete Fourier transform with N entries.
 17. TheOTS according to claim 16, wherein the DFA further comprises an inversediscrete Fourier transformation (IDFT) filter configured to generate, onthe basis of the output discrete Fourier transform, an output block ofsize N.
 18. The OTS according to claim 17, wherein the DFA is configuredto generate an output block of the size L on the basis of the outputblock of the size N, by using an overlap-add or an overlap-save method.19. A method, comprising: receiving a sequence of samples of a digitalinput signal in the time domain in the form of consecutive blocks ofsize L, wherein each block comprises L consecutive samples of thedigital input signal; generating M discrete Fourier transforms of acurrent overlap block and of M−1 delayed versions of the current overlapblock by using M discrete Fourier transform (DFT) filters, wherein thecurrent overlap block is of a size N greater than the size L, eachgenerated discrete Fourier transform is of the size N and comprises Nentries, the current overlap block comprises the samples of a currentblock and the N-L last consecutive samples of a directly previous blockthat was received directly before the current block, and each DFT filterof the M DFT filters is implemented by a fast Fourier transform (FFT)algorithm, of a size Γ smaller than the size N and by an interpolationalgorithm; and filtering, by a compensation filter, the entries of thegenerated M discrete Fourier transforms to generate an output discreteFourier transform with N entries, wherein the compensation filter isimplemented by a delay network and a linear combination algorithm.
 20. Anon-transitory computer readable medium storing executable program codewhich, when executed by a processor, causes the method according toclaim 19 to be performed.